This Document Contains Chapters 6 to 7 Brealey 5CE Solutions to Chapter 6 Note: Unless otherwise stated, assume all bonds have $1,000 face (par) value. 1. a. The coupon payments are fixed at $60 per year. Coupon rate = coupon payment/par value = 60/1000 = 6%, which remains unchanged. b. When the market yield increases, the bond price will fall. The cash flows are discounted at a higher rate. c. At a lower price, the bond’s yield to maturity will be higher. The higher yield to maturity on the bond is commensurate with the higher yields available in the rest of the bond market. d. Current yield = coupon payment/bond price. As coupon payment remains the same and the bond price decreases, the current yield increases. 2. When the bond is selling at a discount, $970 in this case, the yield to maturity is greater than 8%. We know that if the discount rate were 8%, the bond would sell at par. At a price below par, the YTM must exceed the coupon rate. Current yield equals coupon payment/bond price, in this case, 80/970. So, current yield is also greater than 8%. 3. Coupon payment = .08 x 1000 = $80 Current yield = 80/bond price = .07 Therefore, bond price = 80/.07 = $1,142.86 4. Par value is $1000 by assumption. Coupon rate = $80/$1000 = .08 = 8% Current yield = $80/$950 = .0842 = 8.42% Yield to maturity = 9.1185% [Enter in the calculator: N = 6; PV= -950; FV = 1000; PMT = 80] 5. To sell at par, the coupon rate must equal yield to maturity. Since Circular bonds yield 9.1185%, this must be the coupon rate. 5-1 This Document Contains Chapters 6 to 7 6. a. Current yield = annual coupon/price = $80/1,100 = .0727 = 7.27%. b. On the calculator, enter PV = -1100, FV = 1000, n = 10, PMT = 80 Then compute I/Y (or i) and will get YTM = 6.6023%. 7. When the bond is selling at par, its yield to maturity equals its coupon rate. This firm’s bonds are selling at a yield to maturity of 9.25%. So the coupon rate on the new bonds must be 9.25% if they are to sell at par. 8. The current bid yield on the bond was 3.09%. To buy the bond, investors pay the ask price. The investor would pay 108.21 percent of par value. With $1,000 par value, this means paying $1,082.1 to buy a bond. 9. Coupon payment = interest = .05 × 1000 = 50 Capital gain = 1100 – 1000 = 100 Rate of return = interest + capital gain purchase price = 50 + 100 1000 = .15 = 15% 10. Tax on interest received = tax rate × interest = .3 × 50 = 15 After-tax interest received = interest – tax = 50 – 15 = 35 Fast way to calculate: After-tax interest received = (1 – tax rate) × interest = (1 – .3)× 50 = 35 Tax on capital gain = .5 × .3 × 100 = 15 After-tax capital gain = 100 – 15 = 85 Fast way to calculate: After-tax capital gain = (1 – tax rate) × capital gain = (1 – .5×.3)×100 = 85 After-tax rate of return = after-tax interest + after-tax capital gain purchase price = 35 + 85 1000 = .12 = 12% 11. Bond 1 year 1: PMT = 80, FV = 1000, i = 10%, n = 10; Compute PV0 = $877.11 year 2: PMT = 80, FV = l000, i = 10%, n = 9; Compute PV1 = $884.82 Rate of return = 80 + (884.82 - 877.11) 877.11 = .10 = 10% 5-2 Bond 2 year 1: PMT = 120, FV = 1000, i = 10%, n = 10; Compute PV0 = $1122.89 year 2: PMT = 120, FV = l000, i = 10%, n = 9; Compute PV1 =$1115.18 Rate of return = 120 + (1115.18 - 1122.89) 1122.89 = .10 = 10% Both bonds provide the same rate of return. 12. Accrued interest = Coupon payment × number of days from last coupon to purchase date number of days in coupon period = 22.5 × 136 184 = $16.63 Dirty bond price= clean bond price + accrued interest = $990+ $16.63= $1006.63 The quoted clean price is $990. The bond pays semi-annual interest. The last $22.5 coupon was paid on March 1, 2011, and the next coupon will be paid on September 1, 2011. The number of days from the last coupon payment to the purchase date is 136 (from March 1 to July 15) and the total number of days in the coupon period is 184 (from March 1 to September 1). The accrued interest is $16.63, and the total cost of buying one bond is $1006.63. 13. a. If YTM = 8%, price will be $1000. b. Rate of return = interest + capital gain original price = 1100 80 +(1000 −1100) = -0.0182 = -1.82% c. Real return = 1 + nominal interest rate 1 + inflation rate – 1 = 1.03 0.9818 – 1 = –.0468 = – 4.68% 14. a. With a par value of $1000 and a coupon rate of 8%, the bondholder receives 2 payments of $40 per year, for a total of $80 per year. b. Assume it is 9%, compounded semi-annually. Per period rate is 9%/2, or 4.5% Price = 40 × annuity factor (4.5%, 18 periods) + 1000/1.04518 = $939.20 5-3 c. If the yield to maturity is 7%, compounded semi-annually, the bond will sell above par, specifically for $1,065.95: Per period rate is 7%/2 = 3.5% Price = 40 × annuity factor(3.5%, 18 periods) + 1000/1.03518 = $1,065.95 15. On your calculator, set N = 30, FV =1000, PMT = 80. a. Set PV = -900 and compute the interest rate to find that YTM = 8.9708% b. Set PV = -1000 and compute the interest rate to find that YTM = 8%. c. Set PV = -1100 and compute the interest rate to find that YTM = 7.1796% 16. On your calculator, set N=60, FV=1000, PMT=40. a. Set PV = -900 and compute the interest rate to find that the (semiannual) YTM =4.4831%. The bond equivalent yield to maturity is therefore 4.4831 × 2 = 8.9662%. b. Set PV = -1000 and compute the interest rate to find that YTM = 4%. The annualized bond equivalent yield to maturity is therefore 4 × 2= 8%. c. Set PV = -1100 and compute the interest rate to find that YTM = 3.5917%. The annualized bond equivalent yield to maturity is therefore 3.5917 × 2 = 7.1834%. 17. In each case we solve this equation for the missing variable: Price= 1000/(1 + YTM)maturity Price Maturity (years) YTM 300 30.0 4.095% 300 15.64 8.0% 385.54 10.0 10.0% Alternatively the problem can be solved using a financial calculator: Solving the first question: PV = (-)300, PMT = 0, n = 30, FV = 1000, and compute i. 18. PV of perpetuity = coupon payment/rate of return. PV = C/r = 60/.06 = $1000 If the required rate of return is 10%, the bond sells for: 5-4 PV = C/r = 60/.1 = $600 19. Because current yield = .098375, bond price can be solved from: 90/Price = .098375, which implies that price = $914.87. On your calculator, you can now enter: i = 10; PV = (-)914.87; FV = 1000; PMT = 90, and solve for n to find that n =20 years. 20. Assume that the yield to maturity is a stated rate. Thus the per-period rate is 7%/2 or 3.5%. We must solve the following equation: PMT × annuity factor(3.5%, 18 periods) + 1000/(1.035)18 = $1065.95 To solve, use a calculator to find the PMT that makes the PV of the bond cash flows equal to $1065.95. You should find PMT = $40. The coupon rate is 2×40/1000 = 8%. 21. a. Since the bonds were issued at par value the coupon rate equaled the yield to maturity at issue. With a yield to maturity of 7% at issue, the coupon rate must be 7%. The semi-annual coupon payment is 0.07/2 × $1,000 = $35. With 8 years left to maturity 16 payments of semiannual coupons will be made. Now that the current yield to maturity is 15% the per-period discount rate is .15/2 = .075 Now, the price is 35 × Annuity factor(7.5%, 16 periods) + 1000/1.07516 = $634.34 b. The investors pay $634.34 for the bond. They expect to receive the promised coupons plus $800 at maturity. We calculate the yield to maturity based on these expectations: 35 × Annuity factor(i, 16 periods) + 800/(1 + i)16 = $634.34 which can be solved on the calculator to show that i =6.4941%. On an annual basis, this 2×6.4941% or 12.9882% [Calculator enteries: N = 16; PV = -634.34; FV = 800; PMT = 35] 22. a. Today, at a price of 980 and maturity of 10 years, the bond’s yield to maturity is 8.3% (n = 10, PV = (-) 980, PMT = 80, FV = 1000). In one year, at a price of 1100 and remaining maturity of 9 years, the bond’s yield to maturity is 6.4978% (n = 9, PV = (-) 1100, PMT = 80, FV = 1000). b. Rate of return = 980 80 + (1100 − 980) = 20.41% 23. Assume the bond pays an annual coupon. The answer is: 5-5 PV0 = $908.71 (n = 20, PMT = 80, FV = 1000, i = 9) PV1 = $832.70 (n = 19, PMT = 80, FV = 1000, i = 10) Rate of return = 908.71 80 + 832.70 − 908.71 = .4391% If the bond pays coupons semi-annually, the solution becomes more complex. First, decide if the yields are effective annual rates or APRs. Second, make an assumption regarding the rate at which the first (mid-year) coupon payment is reinvested for the second half of the year. Your assumptions will affect the calculated rate of return on the investment. Here is one possible solution: Assume that the yields are APR and the yield changes from 9% to 10% at the end of the year. The bond prices today and one year from today are: PV0 = $907.99 (n = 2 × 20 = 40, PMT = 80/2 = 40, FV = 1000, i = 9/2 = 4.5) PV1 = $831.32 (n = 2 × 19 = 38, PMT = 80/2 = 40, FV = 1000, i = 10/2 = 5) Assuming that the yield doesn’t increase to 10% until the end of year, the $40 mid-year coupon payment is reinvested for half a year at 9%, compounded monthly. Its future value at the end of the year is: $40 × (1.045) = $41.80 and the rate of return on the bond investment is: Rate of return = 907.99 41.80 + 40 + 831.32 − 907.99 = .56% 24. The price of the bond at the end of the year depends on the interest rate at that time. With one year until maturity, the bond price will be $ 1065/(1 + r). a. Price = 1065/1.06 = $1004.72 Return = [65 + (1004.72 – 1000)]/1000 = .06972 = 6.972% b. Price = 1065/1.08 = $986.11 Return = [65+ (986.11 – 1000)]/1000 = .05111 = 5.111% c. Price = 1065/1.10 = $968.18 Return = [65 + (968.18 – 1000)]/1000 = .0332 = 3.32% 25. The bond price is originally $627.73. (On your calculator, input n = 30, PMT = 40, FV =1000, and i = 7%.) After one year, the maturity of the bond will be 29 years and its price will be $553.66. (On your calculator, input n = 29, PMT = 40, FV = 1000, and i = 8%.) The rate of return is therefore [40 + (553.66 – 627.73)]/627.73 = –.054275 = –5.4275%. 5-6 26. a. Annual coupon = .08 × 1000 = $80. Total coupons received after 5 years = 5 × 80 = $400 Total cash flows, after 5 years = 400 + 1000 = $1400 Rate of return = (1400975 )1/5 – 1 = .075 = 7.5% b. Future value of coupons after 5 years = 80 × future value factor(1%, 5 years) = 408.08 Total cash flows, after 5 years = 408.08 + 1000 = $1408.08 Rate of return = (1408.08975 )1/5 – 1 = .0763 = 7.63% c. Future value of coupons after 5 years = 80 × future value factor(8.64%, 5 years) = 475.35 Total cash flows, after 5 years = 475.35 + 1000 = $1475.35 Rate of return = (1475.35975 )1/5 – 1 = .0864 = 8.64% 27. To solve for the rate of return using the YTM method, find the discount rate that makes the original price equal to the present value of the bond’s cash flows: 975 = 80 × annuity factor( YTM, 5 years ) + 1000/(1 + YTM)5 Using the calculator, enter PV = (-)975, n = 5, PMT = 80, FV = 1000 and compute i. You will find i = 8.64%, the same answer we found in 26 (c). 28. a. False. Since a bond's coupon payments and principal are fixed, as interest rates rise, the present value of the bond's future cash flow falls. Hence, the bond price falls. Example: Two-year bond 3% coupon, paid annual. Current YTM = 6% Price = 30 × annuity factor(6%, 2) + 1000/(1 + .06)2 = 945 If rate rises to 7%, the new price is: Price = 30 × annuity factor(7%, 2) + 1000/(1 + .07)2 = 927.68 b. False. If the bond's YMT is greater than its coupon rate, the bond must sell at a discount to make up for the lower coupon rate. For an example, see the bond in a. In both cases, the bond's coupon rate of 3% is less than its YTM and the bond 5-7 sells for less than its $1,000 par value. c. False. With a higher coupon rate, everything else equal, the bond pays more future cash flow and will sell for a higher price. Consider a bond identical to the one in a. but with a 6% coupon rate. With the YTM equal to 6%, the bond will sell for par value, $1,000. This is greater the $945 price of the otherwise identical bond with a 3% coupon rate. d. False. Compare the 3% coupon bond in a with the 6% coupon bond in c. When YTM rises from 6% to 7%, the 3% coupon bond's price falls from $945 to $927.68, a -1.8328% decrease (= (927.68 - 945)/945). The otherwise identical 6% bonds price falls to 981.92 (= 60 × annuity factor(7%, 2) + 1000/(1 + .07)2) when the YTM increases to 7%. This is a -1.808% decrease (= 981.92 - 1000/1000), which is slightly smaller. The prices of bonds with lower coupon rates are more sensitivity to changes in interest rates than bonds with higher coupon rates. e. False. As interest rates rise, the value of bonds fall. A 10 percent, 5 year Canada bond pays $50 of interest semi-annually (= .10/2 × $1,000). If the interest rate is assumed to be compounded semi-annually, the per period rate of 2% (= 4%/2) rises to 2.5% (=5%/2). The bond price changes from: Price = 50 × annuity factor(2%, 2×5) + 1000/(1 + .02)10 = $1,269.48 to: Price = 50 × annuity factor(2.5%, 2×5) + 1000/(1 + .025)10 = $1,218.80 The wealth of the investor falls 4% (=$1,218.80 - $1,269.48/$1,269.48). 29. Internet: Using historical yield-to-maturity data from Bank of Canada Tips: Students will need to read the instructions on how to put the data into a spreadsheet. They will want to save the data in CSV format so that it will be easily moved into the spreadsheet. The data will be automatically put into Excel if you access the website with Internet Explorer. Watch that the headings for the columns of data in your spreadsheet aren’t out of line (we found that the Government of Canada bond yield heading took two columns, displacing the other two headings – the data itself were in the correct columns). Expected results: Long-term Government of Canada bonds have the lowest yield, followed by the yields for the provincial long bonds and then for the corporate bonds. The graph of the yields clearly shows the consistent spreads but also how the level of interest rates varies over time. For an even clearer picture, have the students pick data from 1990 onward. Time Series: Low/High/Average (Accessed November 22, 2008) Date Range: 2002/07 – 2007/06 'V122544=Government of Canada benchmark bond yields - long-term 5-8 'V122517=Average weighted bond yields (Scotia Capital Inc.) - Provincial - long-term 'V122518=Average weighted bond yields (Scotia Capital Inc.) - All corporates - long-term Date V122544 V122517 V122518 Yield spread (Provincial vs. Canada) Yield Spread (Corporate vs. Canada) 2002/07 5.73 6.13 7.19 0.4 1.46 2002/08 5.58 6 6.99 0.42 1.41 2002/09 5.43 5.83 6.84 0.4 1.41 2002/10 5.63 6.05 7.17 0.42 1.54 2002/11 5.58 5.99 6.96 0.41 1.38 2002/12 5.42 5.81 6.73 0.39 1.31 2003/01 5.49 5.92 6.85 0.43 1.36 2003/02 5.46 5.88 6.81 0.42 1.35 2003/03 5.58 6.02 7.06 0.44 1.48 2003/04 5.41 5.82 6.7 0.41 1.29 2003/05 5.12 5.52 6.35 0.4 1.23 2003/06 5.03 5.41 6.22 0.38 1.19 2003/07 5.4 5.7 6.48 0.3 1.08 2003/08 5.44 5.79 6.54 0.35 1.1 2003/09 5.23 5.57 6.29 0.34 1.06 2003/10 5.38 5.73 6.39 0.35 1.01 2003/11 5.29 5.63 6.27 0.34 0.98 2003/12 5.2 5.52 6.07 0.32 0.87 2004/01 5.23 5.5 6.03 0.27 0.8 2004/02 5.09 5.37 5.87 0.28 0.78 2004/03 5.04 5.38 5.85 0.34 0.81 2004/04 5.31 5.66 6.15 0.35 0.84 2004/05 5.32 5.71 6.25 0.39 0.93 2004/06 5.33 5.78 6.36 0.45 1.03 2004/07 5.29 5.76 6.34 0.47 1.05 2004/08 5.15 5.58 6.17 0.43 1.02 2004/09 5.04 5.44 6.05 0.4 1.01 2004/10 5 5.39 5.99 0.39 0.99 2004/11 4.9 5.29 5.88 0.39 0.98 2004/12 4.92 5.3 5.82 0.38 0.9 2005/01 4.74 5.14 5.66 0.4 0.92 2005/02 4.76 5.11 5.62 0.35 0.86 2005/03 4.77 5.21 5.73 0.44 0.96 2005/04 4.59 5.04 5.58 0.45 0.99 2005/05 4.46 4.89 5.46 0.43 1 2005/06 4.29 4.69 5.2 0.4 0.91 2005/07 4.31 4.72 5.25 0.41 0.94 2005/08 4.12 4.52 5.04 0.4 0.92 2005/09 4.21 4.64 5.15 0.43 0.94 2005/10 4.37 4.82 5.34 0.45 0.97 2005/11 4.18 4.67 5.24 0.49 1.06 2005/12 4.02 4.54 5.09 0.52 1.07 5-9 2006/01 4.2 4.71 5.3 0.51 1.1 2006/02 4.15 4.67 5.27 0.52 1.12 2006/03 4.23 4.78 5.37 0.55 1.14 2006/04 4.57 5.07 5.67 0.5 1.1 2006/05 4.5 5.01 5.6 0.51 1.1 2006/06 4.67 5.18 5.81 0.51 1.14 2006/07 4.45 4.96 5.6 0.51 1.15 2006/08 4.2 4.69 5.33 0.49 1.13 2006/09 4.07 4.55 5.18 0.48 1.11 2006/10 4.24 4.7 5.33 0.46 1.09 2006/11 4.02 4.47 5.11 0.45 1.09 2006/12 4.1 4.56 5.18 0.46 1.08 2007/01 4.22 4.66 5.28 0.44 1.06 2007/02 4.09 4.53 5.15 0.44 1.06 2007/03 4.21 4.64 5.27 0.43 1.06 2007/04 4.2 4.64 5.38 0.44 1.18 2007/05 4.39 4.84 5.63 0.45 1.24 2007/06 4.56 5.07 5.82 0.51 1.26 Average Yield Spread of the provincial bonds over the Canada bonds：0.42% Average Yield Spread of the corporate bonds over the Canada bonds： 1.09% Yields Spread 3.5 4 4.5 5 5.5 6 6.5 7 7.5 2002/07 2002/12 2003/05 2003/10 2004/03 2004/08 2005/01 2005/06 2005/11 2006/04 2006/09 2007/02 Date YTM (percent) YTM of long-term Corporate Bonds YTM of long-term Provincial Bonds YTM of long-term Canada Bonds We can see that long-term Government of Canada bonds have the lowest yield over time, followed by the yields for long-term provincial long bonds and then for the corporate bonds. The graph of the yields clearly shows the consistent spreads but also how the level of interest rates varies over time. The result makes sense because YTM of long-term Canada bonds has the lowest risk premium of the three, followed by YTM of the provincial bonds. YTM of long-term corporate bonds has 5-10 larger spreads over Canada bonds because it has much higher default and liquidity risk than Canada Bonds. 30. a. Strips pay no interest, only principal. Assume each bond pays $100 principal on the maturity date Bond Time to Maturity (Years) YTM = (100/Price)1/time to maturity - 1 June 2014 1.583 = (100/96.94)1/1.583 - 1 = .0198 June 2016 3.583 = (100/91.04)1/3.583 - 1 = .02655 June 2019 6.583 = (100/80.58)1/6.583 - 1 = .03334 June 2023 10.583 = (100/65.43)1/10.583 - 1 = .04090 June 2029 16.583 = (100/45.75)1/16.583 - 1 = .04828 b. The term structure (yield curve) is upward sloping. 31. Price of bond today = 40 × PVIFA(5%, 3) + 50 × PVIFA(5%,3) × PVIF(5%,3) + 60 × PVIFA(5%,3)×PVIF(5%,6) + 1000 × PVIF(5%, 9) = 108.93 + 117.62 + 121.93 + 644.61 = $993.09 32. a., b. Price of each bond at different yields to maturity Maturity of bond 4 years 8 years 30 years Yield (%) 7 1033.87 1059.71 1124.09 8 1000.00 1000.00 1000.00 9 967.60 944.65 897.26 Difference between prices (YTM=7% vs YTM=9%) 66.27 115.06 226.83 c. The table shows that prices of longer-term bonds respond with more sensitivity to changes in interest rates. This can be illustrated in a variety of ways. In the table we compare the prices of the bonds at 7 percent and 9 percent yields. When the yield falls from 9 to 7%, the price of the 30-year bond increases $226.83 but the price of the 4-year bond only increases $66.27. Another way to compare the bonds’ sensitivity to changes in the yield is to look at the percentage change in the prices. For example, with an increase in the yield from 8 to 9%, the price of the 4-year bond falls (967.6/1000) –1, or 3.24% but the 30-year bond price falls (897.26/1000) – 1, or 10.27%. 5-11 33. The bond’s yield to maturity will increase from 7.5%, effective annual interest (EAR) to 7.8%, EAR, when the perceived default risk increases. 6 month interest rate equivalent to 7.5% EAR = (1.075)1/2 – 1 = .036822 6 month interest rate equivalent to 7.8% EAR = (1.078)1/2 – 1 = .038268 Price at AA rating = $974.53 (n = 2×10 = 20, PMT = 70/2 = 35, FV =1000, i = 3.6822) Price at A rating = $954.90 (n = 2×10 = 20, PMT = 70/2 = 35, FV =1000, i = 3.8268) The price falls by $19.63 dollars due to the drop in the bond rating and the increase in the required rate of return. 34. Internet: Credit spreads on corporate bonds At www.bondsonline.com/Todays_Market/Corporate_Bond_Spreads.php, the spread for a 10 year A2/A-rated bond was reported to be “95”, meaning 95 basis points (bp) or .95%. The spread for a 10 year B2/B-rated bond was 405 bp or 4.05%. As of August 1,2011 the yield to maturity on a 10 year US Treasury bond was 2.78%. The estimated required rate of return on each corporate bond is: Required rate of return = US treasury bond yield to maturity + credit spread 10 year A2/A-rated bond required rate of return = 2.78% + .95% = 3.73% 10 year B2/B-rated bond required rate of return = 2.78% + 4.05% = 6.83% 35. Internet: Canadian corporate bond yields (As of August 1, 2011) Tips: If you click on “Bond Type” it will sort the bonds by type, making it easier to find a set of corporate bonds. Alternatively, the data in the table can be copied and pasted into Excel and sorted there. If you sort by type and maturity, it is easier to get a group of corporate bonds with similar maturity dates. At www.dbrs.com, type the company name into the search box. If the company is rated, it will be listed. Click on the name and pick the rating of the subordinated debt (or just the lowest rating). Find a Government of Canada bond (CANADA FEDGOV) with a similar maturity date in the bond list. Calculate the yield spread: corporate bond yield – government bond yield and compare the yields and spread with the different ratings. 5-12 Here’s sample of data taken from the globeinvestor.com bond table and assembled into a table in Excel. “Spread” in the final column is calculated as the difference between the corporate bond yield and the corresponding Government of Canada bond. Federal government bonds could not be found with exactly the same maturity date for all corporate bonds. So the Federal bond with closest maturity date was chosen. The BBB rate bonds have the largest spread, between 94 and 132 basis points. By contrast the two AA corporate bonds (Bank of Montreal and Bank of Nova Scotia) are substantially smaller, only about 78 to 91 basis points. CORPORATE BONDS CANADA FEDGOV DBRS Rating Coupon Rate Coupon Freq. Maturity Price Yield Maturity Yield Spread CANADIAN NATURAL RESOURCES BBB(high) 4.95 S 06/01/2015 108.73 2.53 06/01/2015 1.59 0.94 METRO INC BBB 4.98 S 10/15/2015 108.82 2.65 06/01/2015 1.59 1.06 ROGERS COMM BBB 5.8 S 05/26/2016 111.76 3.14 06/01/2016 1.82 1.32 BANK OF MTL AA(low) 4.55 S 08/01/2017 107.96 2.95 06/01/2017 2.04 0.91 BANK OF NOVA SCOTIA AA(low) 3.93 S 04/27/2015 105.43 2.37 06/01/2015 1.59 0.78 36. YTM = 4% Real interest rate = 1 + nominal interest rate = 1.04 - 1 = .0196, or 1.96% 1 + expected rate of inflation 1.02 Real interest rate ≈ nominal interest rate - expected inflation rate = 4% - 2% = 2% 37. The nominal return is 1060/1000, or 6%. The real return is 1.06/(1 + inflation) – 1. a. 1.06/1.02 – 1 = .0392 = 3.92% b. 1.06/1.04 – 1 = .0192 = 1.92% c. 1.06/1.06 – 1 = 0% d. 1.06/1.08 – 1 = – .0185 = –1.85% 38. The principal value of the bond will increase by the inflation rate, and since the coupon is 4% of the principal, it too will rise along with the general level of prices. The total cash flow provided by the bond will be 1000 × (1 + inflation rate) + coupon rate × 1000 × (1 + inflation rate). 5-13 Since the bond is purchased for par value, or $1000, total dollar nominal return is therefore the increase in the principal due to the inflation indexing, plus coupon income: Income = 1000 × inflation rate + coupon rate × 1000 × (1 + inflation rate) Finally, the nominal rate of return = income/1000. a. Nominal return = 20 + 40 × 1.02 1000 = .0608 Real return = 1.0608 1.02 – 1 = .04 b. Nominal return = 40 + 40 × 1.04 1000 = .0816 Real return = 1.0816 1.04 – 1 = .04 c. Nominal return = 60 + 40 × 1.06 1000 = .1024 Real return = 1.1024 1.06 – 1 = .04 d. Nominal return = 80 + 40 × 1.08 1000 = .1232 Real return = 1.1232 1.08 – 1 = .04 39. First year income Second year income a. 40x1.02=$40.80 1040 x 1.022 = $1082.02 b. 40x1.04=$41.60 1040 x 1.042 = $1124.86 c. 40x1.06=$42.40 1040 x 1.062 = $1168.54 d. 40x1.08=$43.20 1040 x 1.082 = $1213.06 40. a. YTM = 5.76% (n=15, PV = (-)1048, PMT=62.5, FV=1000) b. YTC = 6.33% (n=10, PV = (-)1048, PMT=62.5, FV=1100) 41. a. Current price = 1,112.38 (n=6, i=4.8%, PMT=70, FV=1000) b. Current call price = 1,137.35 (n=6, i=4.35%, PMT=70, FV=1000) 42. a. YTM on ABC bond at issue = 5.5% (since sold at par, coupon rate = required rate of return) 10-year Gov't of Canada bond yield at issue = ABC bond YTM - credit spread = 5.5% - .25% = 5.25% Required yield to meet Canada call: = 10-year Gov't of Canada bond yield + .15% = 5.25 + .15% = 5.4% Call price at issue = 1,007.57 (n=10, i=5.4%, PMT=55, FV=1000) 5-14 b. Required yield to call bond = 4.9% + .15% = 5.05% Call price now, 5 years later = 1,019.46 (n=5, i=5.05%, PMT=55, FV=1000) c. Based on new interest rates, the bond price is: Price now, 5 years later = 1,021.65 (n=5, i=5%, PMT=55, FV=1000) Now the current price is greater than the call price. The company can call bonds and reduce its cost of debt. 43. The coupon bond will fall from an initial price of $1000 (when yield to maturity = 8%) to a new price of $897.26 when YTM immediately rises to 9%. This is a 10.27% decline in the bond price. The zero coupon bond will fall from an initial price of 1000 1.0830 = $99.38 to a new price of 1000 1.0930 = $75.37. This is a price decline of 24.16%, far greater than that of the coupon bond. The price of the coupon bond is much less sensitive to the change in yield. It seems to act like a shorter maturity bond. This makes sense: the 8% bond makes many coupon payments, most of which come years before the bond’s maturity date. Each payment may be considered to have its own “maturity date” which suggests that the effective maturity of the bond should be measured as some sort of average of the maturities of all the cash flows paid out by the bond. The zero–coupon bond, by contrast, makes only one payment at the final maturity date. 44. a. Annual after-tax coupon = (1 - .35) × .08 × 1000 = $52. Total coupons received after 5 years = 5 × 52 = $260 Capital gains tax = .5 × .35 × (1000 – 975) = 4.375 After-tax capital gains = 1000 – 975 – 4.375 = 20.625 Total cash flows, after 5 years = 260 + 1000 – 4.375 = $ 1255.625 Rate of return = (1255.625975 )1/5 – 1 = .05189, or 5.189% Note: This can also be answered by first calculating the five-year rate of return and then converting it into a one-year rate of return. This way students can continue to use the coupons + capital gains/original investment approach: Five-year rate of return =after-tax coupons + after-tax capital gain original investment 5-15 = 260 + 20.625 975 = .28782 The one-year rate of return equivalent to the five-year rate of return is: (1 + .28782) 1/5 – 1 = .05189, or 5.189%. b. Future value of coupons after 5 years = (1 – .35) × 80 × future value factor((1–.35)×1%, 5 years) = 263.4 Total cash flows, after 5 years = 263.4 + 1000 – 4.375 = $1259.025 Rate of return = (1259.025975 )1/5 – 1 = .0525 = 5.246% c. Future value of coupons after 5 years = (1 – .35) × 80 × future value factor((1–.35)×8.64%, 5 years) = 290.89 Total cash flows, after 5 years = 290.89 + 1000 – 4.375 = $1286.5 Rate of return = (1286.5975 )1/5 – 1 = .057 = 5.7% 45. The new bonds must be priced to have a yield to maturity of 5% + 1.5% = 6.5%. To sell at par, the coupon rate on the new bonds must be set at 6.5%. 46. Expected results: Students should be able to see some evidence supporting the difference in the bond ratings of these two companies. BCE, Inc. provides wire line and wireless communications services, Internet access, data services, and video services in Canada. BCE has DBRS rating of BBB(high) Agrium, Inc. produces and markets agricultural nutrients, industrial products, and specialty products worldwide. The company has DBRS Issuer credit rating: BBB BCE: Times interest earned= EBIT interest payment = 5.67 BCE: Debt/Equity = 61.49% AGU: Times interest earned= EBIT interest payment = 10.31 AGU: Debt/Equity= 39.61% Agrium has a higher times interest earned ratio of 10.31 while BCE’s times interest earned is 5.67. Thus, Agrium has greater ability to make its interest payment than BCE. 5-16 BCE’s indebtedness is higher than AGU because it has higher debt to equity ratio than AGU. However, both ratios contradicts BCE’s higher credit rating. When providing a credit rating to a firm, each company’s business risk is evaluated in addition to the financial risks. Evidently, Agrium’s higher business risks resulted in a slightly lower credit rating than BCE even though its times interest earned is higher and indebtedness is lower. Appendix 6A Solutions 6A.1 a. Equation 6B.4: (1 + rn)n = (1 + rn−1)n−1 × (1 + fn) rn = spot interest rate for n year investment rn-1 = spot interest rate for n-1 year investment fn = forward interest rate for year n Rearrange equation 6B.4 to solve for the forward rate: fn = (1 + rn)n - 1 (1 + rn−1)n−1 Year 2 forward rate = (1+.02)2 -1 = 0.2745 = 2.745% (1.0126) Year 3 forward rate = (1+.0247)3 -1 = .03417 = 3.417% (1.02)2 Year 4 forward rate = (1+.0279)4 -1 = .03756 = 3.756% (1.0247)3 Year 5 forward rate = (1+.0302)5 -1 = .03945 = 3.945% (1.0279)4 b. To calculate the bond prices use the yield to maturity that corresponds to the payment date for the bonds: (i) 5%, 2-Year bond: Annual coupon payment = .05 x 1000 = $50 Price today = = 1,058.61 (ii) 5%, 5-Year bond: Annual coupon payment = .05 x 1000 = $50 Price today = = 1,093.56 (iii) 10%, 5-Year bond: Annual coupon payment = .10 x 1000 = $100 5-17 Price today = = 1,325.34 c. Using calculator to calculate yield to maturity: 5%, 2-Year bond: PMT = 50, N= 2, FV = 1000, PV = -1,058.61 YTM (I/Y) = 1.98% 5%, 5-Year bond: PMT = 50, N= 5, FV = 1000, PV = -1,093.56 YTM (I/Y) = 2.96% 10%, 5-Year bond: PMT = 100, N= 5, FV = 1000, PV = -1,325.34 YTM (I/Y) = 2.91% d. The 10% 5-year bond yield to maturity is slightly lower than the 5% 5-year coupon bond. The purchase price of the 10% coupon 5-year bond is higher than the purchase price of the 5% coupon 5-year bond. Although both bond prices were calculated using the same discount rates you have to pay more to buy the bond that pays higher coupon payments each year. But the bond that pays the higher coupon payment has more payments earlier. Since the yield to maturity is effectively an average of the future interest rates that fact that the 10% bond pays more earlier, when rates were lower, the effective interest rate on the investment (the yield to maturity) is slightly lower for the 10% bond! 6A.2 a. The forward rates are higher each year. If the expectations theory is correct, the forward rates are also the expected future interest rates. The expected future interest rates indicate that interest rates are expected to increase over time! b. With liquidity-preference, longer-term bonds earn higher return to compensate investors for the liquidity risk. So the spot rate on a longer term bond includes both expectation of future interest rates and also liquidity risk premium. So, unfortunately we can’t be sure that the future rates are only due to expectation of future interest rates. 6A.3 a. Since each bond pays zero coupons (strip bonds) the yield to maturity for a bond maturing in n years from today can be calculated as: YTM = maturity payment 1/n current price The 2014 strip bond matures in 1 year from 2013 so n = 1 YTM on 2014 strip bond = (1000/988.53) -1 = .0116 = 1.16% The 2015 strip bond matures in 2 year from 2013 so n = 2 YTM on 2015 strip bond = (1000/969.15)1/2 -1 = .0158 = 1.58% The 2016 strip bond matures in 3 year from 2013 so n = 3 YTM on 2015 strip bond = (1000/945.5)1/3 -1 = .0189 = 1.89% 5-18 b. The yield to maturity for the 2014 bond, 1.16%, is the one year interest rate as of June 2013, r2013. Now the yield to maturity for the 2015 bond (YTM2015), 1.58%, reflects the 2013 interest rate and also the forward interest rate as of June 2014, f2014. To calculate the June 2014 forward rate you can use this version of Equation 6B.2: (1 + YTM2015)2 = (1+ r2013) x (1+ f2014) So: f2014 = [(1 + YTM2015)2 / (1+ r2013)] -1 = [(1.0158)2 /(1.0116)] – 1 = .0200 = 2.00% The forward interest rate as of June 2015, f2015, is calculated with the yield to maturity on the 2015 bond (YTM2015), and the yield to maturity on the 2016 bond (YTM2016). Buying the 2016 bond is making a 3 year investment. Here’s the formula: (1 + YTM2016)3 = (1 + YTM2015)2 x (1+ f2015) So: f2015 = [(1 + YTM2016)3 / (1 + YTM2015)2] -1 = [(1.0189)3/(1.0158)2]-1 = .0251 = 2.51% Note: Because there is no bond maturing in 2017 the forward rate for June 2016 cannot be calculated. c. Assume that this is a bond with $1000 face value and pays annual coupons in June of each year. So the annual coupon is .05 x 1000 = $50. If you discount at the yield to maturity for bonds maturing at each of payment dates the price of the bond at June 2013 is: Price today = = 1,090.53 Note: if you use the forward rates to discount the coupons and principal and carry all the decimal places you would get the same answer. Rather than doing the calculation we will show you that the formula for discounting using the forward rates is equivalent to discounting using the yields to maturity. The present value of the $50 payment at June 2015 was calculated above as 50/(1+ YMT2015)2 = 50/(1.0158)2 It could also be calculated by first discounting by the 2014 forward rate and then by the 2013 interest rate: PV of 2014 interest payment = (1+50f2014) x (1+ 1r2013) = (1+ f2014)50x (1+ r2013) But (1 + YTM2015)2 = (1+ r2013) x (1+ f2014) = So: = (1+ f2014)50x (1+ r2013) 5-19 Also if the payment at the end of 2015 were discounted by the forward rates and the 2013 interest rate the answer would be the same as discounting using the 2015 yield to maturity: PV of 2015 interest payment = 50+1000 (1+ f2015) x 1 (1+ f2014) x 1 (1+ r2013) = (1+ f2015)x (1+50f2014) x (1+ r2013) But (1 + YTM2016)3 = (1 + YTM2015)2 x (1+ f2015) and (1 + YTM2015)2 = (1+ r2013) x (1+ f2014) So: (1 + YTM2016)3 = (1+ r2013) x (1+ f2014)x (1+ f2015) So the price of the bond if all of the cash flows are discounted using all of the forward rates will be exactly the same as discounting using the yield to maturities. You can do the calculations but you must carry of the decimal places to make the numbers exactly the same. 6A.4 Assuming the expectations theory the upward sloping yield curve implies that future annual interest rates will be higher than the current interest rate. If the company only needs money to borrow money for a short period, then it will be cheaper to borrow short term than long term. Now, the other issue is that liquidity premium. So if you borrow long term you must pay the liquidity risk premium. So even if future rates aren’t much higher than today if liquidity is relevant, borrowing short term again and again can be cheaper than borrowing long term. However, there is one other issue for a company. If they need to borrow for a long period of time but borrow short term, they need to repeatedly negotiate loans, every time the loan matures. Then if they get into financial trouble or there is an unanticipated financial crisis, they might not be able to get the new loan in the future. So, borrowing long term can be less risky but possibly more expensive. 5-20 Brealey 5CE Solutions to Chapter 7 1. No. The dividend discount model allows for the fact that firms may not currently pay dividends. As the market matures, and Research in Motion’s growth opportunities moderate, investors may justifiably believe that Research in Motion will enjoy high future earnings and will pay dividends then. The stock price today can still reflect the present value of the expected per share stream of dividends. 2. Dividend yield = Expected dividend/Price = DIV1/P0 So: P0 = DIV1/dividend yield P0 = $2.4/.08 = $30 3. a. The typical preferred stock pays a level perpetuity of dividends. The expected dividend next year is the same as this year’s dividend, $8. Thus the dividend growth rate is zero and the price today is: P0 = D1/r = 8/.12 = $66.67 b. The expected dividend in two years is this year’s dividend, $8. P1= D2/r = 8/.12 = $66.67 c. Dividend yield = $8/$66.67 = .12 =12% Expected capital gains = 0 Expected rate of return = 12% 4. r = DIV1/P0 + g = 8% + 5% = 13% 5. The value of a common stock equals the present value of dividends received out to the investment horizon, plus the present value of the forecast stock price at the horizon. But the stock price at the horizon date depends on expectations of dividends from that date forward. So even if an investor plans to hold a stock for only a year for two, the price ultimately received from another investor depends on dividends to be paid after the date of purchase. Therefore, the stock’s present value is the same for investors with different time horizons. 6. a. P0 = DIV1 r − g ⇒ r = DIV1 P0 + g r = $3/$30+ .04 = .14 = 14% b. P0 = 3.00/(.165 − .04) = $24 7-1 7. The dividend yield is defined as the annual dividend (or the annualized current dividend) divided by the current price. The current annual dividend is ($2 × 4) = $8 and the dividend yield is: DIV1/P0 = .048 ⇒ $8/ P0 = .048 ⇒ P0 = $8/.048 = $166.7 To work with the quarterly dividend, divide the dividend yield by 4 and repeat the above steps: Quarterly DIV/P0 = .048/4 = .012 ⇒ $2/ P0 = .012 ⇒ P0 = $2/.012 = $166.7 8. Weak, semi-strong, strong, fundamental, technical 9. True. The search for information and insightful analysis is what makes investor assessments of stock values as reliable as possible. Since the rewards accrue to the investors who uncover relevant information before it is reflected in stock prices, competition among these investors means that there is always an active search on for mispriced stocks. 10. Two such behavioral biases are alluded to in the text. a.) Attitudes toward risk: Psychologists have observed the tendency of investors to be particularly averse to the possibility of incurring losses. Consequently, when investors sustain a loss they tend to exercise excessive conservatism in their subsequent investment decisions to avoid worsening their deficit. Conversely, when investors have secured a gain they are more eager to take on risky bets because they take comfort in their cushion of profit. This behavioral bias has been implicated as a cause of stock market “bubbles”: a phenomenon where market prices far exceed intrinsic firm values. b.) Beliefs about probabilities: This bias is twofold. Firstly, investors tend to look at previous periods and assume that future market fluctuations will follow suit. This is incorrect since stock prices follow a random walk and ought to be independent of previous market movements. To further compound their error, investors tend to be myopic and place greater emphasis on recent market developments; as such, they overlook valuable information from the distant past (i.e. market reactions to long-term economic cycles). Secondly, investors tend to fall prey to overconfidence. When investors profit in the market they wrongly attribute their success to skill rather than mere luck. Such excessive optimism may cause prices to greatly deviate from intrinsic share values. 7-2 11. a. DIV1 = $1 × 1.04 = $1.04 DIV2 = $1 × 1.042 = $1.0816 DIV3 = $1 × 1.043 = $1.1249 b. P0 = DIV1/(r − g) = .121.04−.04 = $13 c. P3 = DIV4/(r − g) = 1.1249 × 1.04 .12 − .04 = $14.6237 Note: the number of decimal places carried affects the solution. If you used this formula the answer would be: P3 = DIV4/(r − g) = .12 − .04 1x(1.04)4 = $14.623232, about $14.6232 d. Your payments are: Year 1 Year 2 Year 3 DIV 1.04 1.0816 1.1249 Sales Price 14.6237 Total cash flow 1.04 1.0816 15.7486 PV of cash flow .9286 .8622 11.2095 Sum of PV = $13.00, the same as your answer to (b). 12. Dividend growth rate, g = return on equity × plowback ratio: g = .15 × .40 = .06 r = DIV1 P0 + g = 4 40 + .06 = .16 = 16% 13. a. P0 = DIV1 r − g = (3 x 1.05) / (0.15 - 0.05) = $31.50 b. P0 = (3 x 1.05) / (0.12 – 0.05) = $45 The lower discount rate makes the present value of future dividends higher, raising the value of the stock. c. P0 = (3 x (1 – .05) / (.15 – (-.05)) = $14.25 7-3 The price is the present value of the future dividends before they are eroded by the negative growth rate. 14. r = DIV1 P0 + g ⇒ g = r - DIV1 P0 = .14 – 5 50 = .04 = 4%. 15. a. r = DIV1 P0 + g = 27 1.64(1.03) + .03 = .0926 = 9.26%. b. If r = .10, then .10 = 1.64(1.03)/27 + g. So g = .0374 = 3.74%. c. g = Return on equity × plowback ratio 5% = Return on equity × .4 Return on equity = g plowback ratio = .05 .4 = 12.5% 16. P0 = DIV1/(r − g) = [$.50 x 1.06 /(.12 - .06)] / [(1 + .12) ^ 4] = $5.61 17. a. P0 = DIV1/(r − g) = 3 / [.15 – (−.10)] = 3/.25 = $12 b. P1 = DIV2/(r − g) = [3 x (1 − .10)]/ [.15 – (−.10)]= 2.7/.25 = $10.80 c. Rate of return = DIV1 + Capital Gain P0 = 12 3 + (10.80 − 12) = .150 = 15.0% d. “Bad” companies may be declining, but if the stock price already reflects this fact, the investor still can earn a fair rate of return, as we saw in part c. 18. a. (i) reinvest 0% of earnings: g = 0 and DIV1 = $5: P0 = DIV1 r − g = $5 .15 − 0 = $33.33 (ii) reinvest 40%: g = 15% × .40 = 6% and DIV1 = $5 × (1 – .40) = $3: P0 = $3/(.15 – .06) = $33.33 (iii) reinvest 60%: g = 15% × .60 = 9% and DIV1 = $5 × (1 – .60) = $2: 7-4 P0 = 2/(.15 – .09) = $33.33 7-5 b. (i) reinvest 0%: P0 = 5/(.15 – 0) = $33.33 PVGO = $0 (ii) reinvest 40%: P0 = .15 − 3 (.2 × .40) = $42.86 PVGO = $42.86 – $33.33 = $9.53 (iii) reinvest 60%: P0 = .15 − 2 (.2 × .60) = $66.67 PVGO = $66.67 – $33.33 = $33.34 c. In part (a), the return on reinvested earnings is equal to the discount rate. Therefore, the NPV of the firm’s new projects is zero, and PVGO is zero in all cases, regardless of the reinvestment rate. While higher reinvestment results in higher growth rates, it does not result in a higher value of growth opportunities. This example illustrates that there is a difference between growth and growth opportunities. In part (b), the return on reinvested earnings is greater than the discount rate. Therefore, the NPV of the firm’s new projects is positive, and PVGO is positive. PVGO is higher when the reinvestment rate is higher in this case, since the firm is taking greater advantage of its opportunities to invest in positive NPV projects. 19. Stock exchange information Example information: Click on the pin in the Atlantic Ocean to get the Bermuda Stock Exchange: 2007 Listed companies: 53 Types of securities traded: equity and debt The website has limited current stock price information, unlike the Toronto Stock Exchange. Click on the daily trade report and see how variable is trading volumes. 20. Hollywood stock exchange Expected result: A fun experience trading securities. Students won’t have trouble figuring out how to set up an account. 7-6 21. P0 = DIV1/(r − g) where g = sustainable growth rate = plowback ratio x ROE So: P0 = DIV1/(r − (plowback ratio x ROE)) Solve for ROE: r − (plowback ratio x ROE) = DIV1/ P0 ⇒ plowback ratio x ROE = r – (DIV1/ P0) ⇒ ROE =[ r – (DIV1/ P0) / plowback ratio The information for Computer Corp: P0= $50, DIV1= $2.50, r = 15%, plowback ratio = .60 So: $50 = $2.50 / (0.15 – (.60 x ROE)) Computer Corp ROE: ROE =[ r – (DIV1/ P0) / plowback ratio ROE = [ 0.15– ($2.5/$50)] / .60 = 0.1667 = 16.67 % 22. a. P0 = 1 1.10 + 1.25 (1.10)2 + 1.50 + 20 (1.10)3 = 18.10 b. DIV1/P0 = $1/18.10 = .0552 = 5.52% 23. Stock A Stock B a. Payout ratio $1/$2 = .50 $1/$1.50 = .67 b. g = ROE × plowback 15% × .5 = 7.5% 10% × .333 = 3.33% c. Price = DIV1/(r − g) $1 × 1.075 .15 − .075 = $14.33 $1 × 1.0333 .15 − .0333 = $8.85 Note: We interpret “recent” to mean in the past. The current stock price depends on future dividends – so the next dividend must be 1 + g times higher. 24. a. ROE × plowback ratio = 20% × .3 = 6% b. E = $2, plowback ratio = .3, r = .12, g = .06 ⇒ P0 = 2 × (1 − .3) .12 − .06 = $23.33 c. No-growth value = E/r = $2/.12 = $16.67 PVGO = P0 − no-growth value = $23.33 − $16.67 = $6.66 d. P/E = 23.33/2 = 11.665 e. If all earnings were paid as dividends, price would equal the no-growth value, $16.67, and P/E would be 16.67/2 = 8.335. f. High P/E ratios reflect expectations of high PVGO. 7-7 25. a. $2.40 / (0.12 – 0.04)=$30 b. No-growth value = E/r = $3.10/.12 = $25.83 PVGO = P0 − no-growth value = $30 − $25.83 = $4.17 26. a. Earnings = DIV1 = $4. Growth rate g = 0. P0 = .12$4− 0 = $33.33 P/E = 33.33/4 = 8.33 b. If r = .10, P0 = 4 .10 = 40, and P/E increases to 40/4 = 10 A decrease in the required rate of return, holding dividends constant, raises the stock price and the P/E ratio. 27. a. Plowback ratio = 0 implies DIV1 = $3 and g = 0. Therefore, P0 = .103− 0 = $30 and the P/E ratio is 30/3 = 10. b. Plowback ratio = .40 implies DIV1 = $3(1 – .40) = $1.80, and g = 10% × .40 = 4%. Therefore P0 = $1.80/(.10 – .04) = $30 and the P/E ratio is 30/3 = 10. c. Plowback ratio = .80 implies DIV1 = $3(1 – .80) = $.60, and g = 10% × .80 = 8%. Therefore P0 = $.60/(.10 – .08) = $30 and the P/E ratio is 30/3 = 10. Regardless of the plowback ratio, the stock price = $30 because all projects offer return on equity just equal to the opportunity cost of capital. 28. a. P0 = DIV1/(r − g) = $5/(.10 – .06) = $125 b. If Trend line followed a zero-plowback strategy, it could pay a perpetual dividend of $8. Its value would be $8/.10 = $80, and therefore, the value of assets in place is $80. The remainder of its value must be due to growth opportunities, so PVGO = $125 – $80 = $45. 7-8 29. a. g = 20% × .30 = 6% DIV1 = $2(1 – .30) = $1.40 P0 = DIV1/(r − g) = $1.40/(.12 − .06) = $23.33 P/E = 23.33/2 = 11.665 b. If the plowback ratio is reduced to .20, g = 20% × .20 = 4% DIV1 = $2(1 – .20) = $1.60 P0 = DIV1/(r − g) = $1.60/(.12 – .04) = 20 P/E = 20/2 = 10 P/E falls because the firm’s value of growth opportunities is now lower: It takes less advantage of its attractive investment opportunities. c. If the plowback ratio = 0, g = 0, and DIV1 = $2, P0 = $2/.12 = 16.67 and E/P = 2/16.67 = .12 30. a. DIV1 = 2.00 PV = 2/1.10 = 1.818 DIV2 = 2(1.20) = 2.40 PV = 2.40/1.102 = 1.983 DIV3 = 2(1.20)2 = 2.88 PV = 2.88/1.103 = 2.164 b. This could not continue indefinitely. If it did, the stock would be worth an infinite amount. Another way to think about the feasible perpetual growth rate is to compare the company’s growth rate with the growth rate of the economy. The economy grows about 3% a year. To grow faster than the economy as a whole is feasible when the company is small. However, to continue to grow at 20%, the company must take over other companies and eventually become the entire economy. But in the long run, it still can only grow as quickly as the entire economy. So it is impossible to grow at 20% in perpetuity. Think about Microsoft – it has had phenomenal growth partly by acquiring other companies and partly by growing its own businesses. However, even if it were to own all of the companies in the world, eventually its growth rate would fall to the growth rate of the world economy. We are assuming that Bill Gates is not able to successfully market his software to still to be discovered alien worlds!! Finally, note too that the constant dividend growth model fails when the assumed perpetual growth rate is greater than the discount rate. 7-9 31. a. Book value of equity= $100 million First year earnings = Equity × return on equity = $100 million × .24 = $24 million Dividends = Earnings × (1 – plowback ratio) =24 × (1-.50) = $12 million g = return on equity × plowback ratio = .24 × .50 = .12 Market value = $12 million .15 − .12 = $400 million Market-to-book ratio = $400/$100 = 4 b. Now g falls to .10 × .50 = .05, first year earnings decline to $10 million (=$100 million × .1), and dividends decline to $5 million (=$10 million × .5). Market value = $5 million .15 − .05 = $50 million Market-to-book ratio = ½ This makes sense, because the firm now earns less than the required rate of return on its investments. Its project is worth less than it costs. 32. 2010 Earnings per share = $3.38 2010 Dividend per share = $1.70 Current dividend payout ratio = dividend per share/earnings per share = $1.70/$3.380 = 0.503 From 2011 to 2015: earnings expected to growth 7% a year and dividend payout ratio expected to stay the same, so dividends expected to grow 7% a year Forecasted dividends per share: 2011 dividend = 1.07 x 1.7 = 1.819 2012 dividend = 1.07 x 1.819 = 1.946. The rest of them are in this chart, using Excel: 2011 2012 2013 2014 2015 Forecasted Dividend 1.819 1.946 2.083 2.228 2.384 After 2015: Dividend growth rate = investment rate of return x plowback ratio investment rate of return = cost of capital = 8.9% plowback ratio = 1- dividend payout ratio = 1 - .503 = .497 Growth rate = .089 x .497 = 0.0442 2016 Dividend = 2015 dividend x growth rate = 2.384 x 1.0442 = 2.489 7-10 Estimated stock price as of the beginning of 2016 (end of 2015) = 2.489/(.089-.0442) = 55.558 Estimate stock price as of the beginning of 2011 = [1.819/1.089] + [1.946/1.0892] + [2.083/1.0893] + [2.228/1.0894] + [2.384/1.0895] + [55.558]/1.0895 = $44.34 Note if you use Excel to solve the problem more decimal places are maintained and estimated 2011 stock price is $44.38 33. P0 = 2 1.12 + 2.50 1.122 + 18 1.123 = $16.59 34. a. DIV1 = $2 × 1.20 = $2.40 b. DIV1 = $2.40 DIV2 = $2.88 DIV3 = $3.456 P3 = 3.456 × 1.04 .15 − .04 = $32.675 P0 = 2.40 1.15 + 2.88 1.152 + 3.456 + 32.675 1.153 = $28.02 c. P1 = 2.88 1.15 + 3.456 + 32.675 1.152 = $29.825 d. Capital gain = P1 − P0 = $29.825 − $28.02 = 1.805 r = 2.40 + 1.805 28.02 = .15 = 15% 35. a. Note: If students carry at least 4 decimal places, the results will be clearer. Also, it is easier to solve the prices in reverse order. DIV1 = $.5 DIV2 = $.5 DIV3 = $.5 DIV4 = $.5 × 1.04 = $.52 DIV5 = $.5 × 1.042 = $.5408 P4 = DIV5 r − g = $.5408 .11 − .04 = $7.7257 P3 = DIV4 r − g = $.52 .11 − .04 = $7.4286 7-11 P2 = DIV3 r − g = $.5 .11 − .04 = $7.1429 P1 = DIV2 + P2 r − g = $.5 + 7.1429 1 + .11 = $6.8855 P0 = DIV1 + P1 r − g = $.5 + $6.8855 1 + .11 = $6.6536 b. Year 0 Dividend yield = D1 P0 = .5 6.6536 = .07515 Capital gains yield = P1 -P0 P0 = 6.8855 - 6.6536 6.6536 = .03485 Dividend yield + capital gains yield = .07515 + .03485 = .11 Year 1 Dividend yield = D2 P1 = .5 6.8855 = .07262 Capital gains yield = P2 -P1 P1 = 7.1429 - 6.8855 6.8855 = .03738 Dividend yield + capital gains yield = .07262 + .03738 = .11 Year 2 Dividend yield = D3 P2 = .5 7.1429 = .0700 Capital gains yield = P3 -P2 P2 = 7.4286 - 7.1429 7.1429 = .0400 Dividend yield + capital gains yield = .07 + .04 = .11 Year 3 Dividend yield = D4 P3 = .52 7.4286 = .0700 Capital gains yield = P4 -P3 P3 = 7.7257-7.4286 7.4286 = .0400 Dividend yield + capital gains yield = .07 + .04 = .11 Yes, each year the sum of the dividend yield and the capital gains yield equal 11 7-12 percent, the required rate of return. Once the company hits constant growth rate of 4 percent, both the dividend yield and the capital gains yield also become constant. 36. DIV1 = dividend payout × earnings1 = .4 × $3 = $1.2 DIV2 = dividend payout × earnings2 = .4 × $3 × 1.1 = $1.32 DIV3 = dividend payout × earnings3 = .4 × $3 × 1.12 = $1.452 DIV4 = dividend payout × earnings4 = .4 × $3 × 1.13 = $1.5972 DIV5 = dividend payout × earnings5 = .4 × $3 × 1.14 = $1.75692 P0 = 1.2 1.15 + 1.32 1.152 + 1.452 1.153 + 1.5972 1.154 + 1.75692 .15 - .05 × 1 1.154 = $13.95 37. a. BCE Series T is listed as BCE.PR.T-T. Click on the actual share name and go to the page describing the preferred share. The dividend will be listed as $1.13 due to rounding. Suppose the traded price was $22.97 (closing price on Aug 19, 2011). Then the expected rate of return, or yield to maturity is: r = DIV1 P0 = $1.13/$22.97 =.049 or 4.9% . b. CIBC preferred shares Expected results: Students will quickly learn that the www.globeinvestor.com does not do a good job on preferred shares. The dividend payouts are rounded to 2 digits and hence do not match well the actual dividends listed on the CIBC Web site. Example: From the CIBC website, details on CM.PR.A are found: Ticker Symbol CUSIP Number Quarterly Dividend Initial Dividend Amount Initial Payment Date CM.PR.P 136069721 $0.343750 $0.331507 Apr 28/98 At Globeinvestor.com on the preferred stock list, click on CM.PR.A and see details on the preferred stock. The annual dividend is listed as as $1.38. The actual annual dividend is $1.375, 4 times the quarterly dividend of $.343750. If the traded stock price is $26.65 (closing price on Aug 19, 2011), the expected annual rate of return on preferred share is $1.375/$26.65 = 0.0516 or 5.16%. 38. Expected Results: MB– T is Mega Brands Inc. and SRF-T is Sun-Rype Foods. Emphasize to the students that they should describe the businesses of these companies in their own words. Encourage students to think about how each company presents its products. Students will look at the investor pages to see the 7-13 information they provide. Example: Mega Brands manufactures and markets over 60 different toys, many of which are molded plastic. Mega Blocks is a competitor to LEGO. They also have paper products such as puzzles and crafts. The investor page provides access to financial reports, such as quarterly and annual reports, press releases, audio broadcasts of analyst conference call and a link to their stock listing at the Toronto Stock Exchange. 39. Before-tax rate of return: = DIV1 + Capital Gain P0 = 25 .7 + (26.25 − 25) = .078 = 7.8% After-tax rate of return: = After-tax DIV1 + After-tax Capital Gain P0 After-tax Dividend = (1- dividend tax rate) x Dividend After-tax Capital gain = (1- capital gain tax rate) × Capital gain Dividend tax rate = 30% Capital gain tax rate = 50% of personal tax rate = .5 × 40% = 20% After-tax rate of return: = (1 - .3) ×.7 + (1 - .2) × (26.25 − 25) 25 = .0596 = 5.96% 40. a. An individual can do crazy things, but still not affect the efficiency of markets. An irrational person can give assets away for free or offer to pay twice the market value. However, when the person’s supply of assets or money runs out, the price will adjust back to its prior level (assuming there is no new, relevant information released by his/her action). If you are lucky enough to trade with such a person you will receive a positive gain at that investor’s expense. You had better not count on this happening very often though. Fortunately, an efficient market protects irrational investors in cases less extreme than the above. Even if they trade in the market in an “irrational” manner, they can be assured of getting a fair price since the price reflects all information. b. Yes, and how many people have dropped a bundle? Or more to the point, how many people have made a bundle only to lose it later? People can be lucky and some people can be very lucky; efficient markets do not preclude this possibility. Furthermore, how much risk did they take? You expect to earn a higher return if you take on more market (beta) risk. c. Investor psychology is a slippery concept, more often than not used to explain price movements which the individual invoking it cannot personally explain. 7-14 Even if it exists, is there any way to make money from it? If investor psychology drives up the price one day, will it do so the next day also? Or will the price drop to a “true” level? Almost no one can tell you beforehand what “investor psychology” will do. Theories based on it have no content. 41. There are several thousand mutual funds in Canada and the United States. With so many professional managers, it is no surprise that some managers will demonstrate brilliant performance over various periods of time. As an analogy, consider a contest in which 10,000 people flip a coin 20 times. It would not surprise you if someone managed to flip heads 18 out of 20 times. But it would be surprising if he could repeat that performance. Similarly, while many investors have shown excellent performance over relatively short time horizons, and have received favourable publicity for their work, far fewer have demonstrated consistency over long periods. 42. If the firm is stable and well run, its price will reflect this information, and the stock may not be a bargain. There is a difference between a “good company” and a “good stock.” The best buys in the stock market are not necessarily the best firms; instead, you want firms that are better than anyone else realizes. When the market catches up to your assessment and prices adjust, you will profit. 43. Remember the first lesson of market efficiency: The market has no memory. Just because long-term interest rates are high relative to past levels doesn’t mean they won’t go higher still. Unless you have special information indicating that long-term rates are too high, issuing long-term bonds should be a zero-NPV transaction. So should issuing short-term debt or common stock. 44. The stock price will fall. The original price would reflect an anticipation of a 25% increase in earnings. The actual increase is a disappointment compared to original expectations. 45. a. DIV1 = $4, and g = 4% Expected return = DIV1/P0 + g = 4/100 + 4% = 8% b. Since DIV1 = Earnings × (1 – plowback ratio), Earnings = DIV1/(1 – plowback ratio) = 4/(1 – .4) = $6.667 If the discount rate is 8% (the expected return on the stock), then the no-growth value of the stock is 6.667/.08 = $83.34. Therefore PVGO =$100 – $83.34 = $16.66 c. For the first 5 years, g = 10% × .8 = 8%. Thereafter, g = 10% × .4 = 4% Year 1 2 3 4 5 6 . . . Earnings 6.67 7.20 7.78 8.40 9.07 9.80 plowback .80 .80 .80 .80 .80 .40 DIV 1.33 1.44 1.56 1.68 1.81 5.88 g .08 .08 .08 .08 .08 .04 7-15 After year 6, the plowback ratio falls to .4 and the growth rate falls to 4 percent. [We assume g = 8% in year 5 (i.e., from t = 5 to t = 6), since the plowback ratio in year 5 is still high at b = .80. Notice the big jump in the dividend when the plowback ratio falls.] By year 6, the firm enters a steady- growth phase, and the constant-growth dividend discount model can be used to value the stock. The stock price in year 6 will be P6 = D6(1 + g) k − g = 5.88(1.04) .08 − .04 = $152.88 P0 = 1.33 1.08 + 1.44 1.082 + 1.56 1.083 + 1.68 1.084 + 1.81 1.085 + 5.88 + 152.88 1.086 = $106.22 46. a. DIV1 = 1.00 × 1.20 = $1.20 DIV2 = 1.00 × (1.20)2 = $1.44 DIV3 = 1.00 × (1.20)3 = $1.728 DIV4 = 1.00 × (1.20)4 = $2.0736 b. P4 = DIV5/(r − g) = DIV4(1 + g)/(r − g) = 2.0736(1.05)/(.10 – .05) = $43.55 c. P0 = 1.20 1.10 + 1.44 (1.10)2 + 1.728 (1.10)3 + 2.074 + 43.55 (1.10)4 = $34.74 d. DIV1/P0 = 1.20/34.74 = .0345 = 3.45% e. Next year the price will be: 1.44 1.10 + 1.728 (1.10)2 + 2.074 + 43.55 (1.10)3 = $37.02 f. return = DIV1 + Capital Gain P0 = 1.20 + (37.02 − 34.74) 34.74 = .10 The expected return equals the discount rate (as it should if the stock is fairly priced). 7-16 47. Before-tax rate of return = DIV1 + Capital Gain P0 = 50 2 + (53 − 50) = .1 = 10% After-tax dividend: Grossed up dividend = 1.25 × 2 = 2.50 Gross federal tax = .22 × 2.50 = .55 Federal tax credit = .1333 × 2.50 = .3333 Net federal tax = .55 - .3333 = .2167 Gross provincial tax = .119 × 2.50 = .2975 Provincial tax credit = .066 × 2.50 = .165 Net provincial tax = .2975 - .165 = .1325 Total dividend tax = .2167 + .1325 = .3492 After-tax dividend = 2 – .3492 = 1.6508 Capital gains tax = .5 × (.22 + .119) × (53 – 50) = .5085 After-tax capital gains = 53 – 50 – .5085 = 2.4915 After-tax rate of return = After-tax DIV1 + After-tax Capital Gain P0 = 1.6508 + 2.4915 50 = .0828 = 8.28% 48. Assume taxes do not change. We make the easiest reinvestment assumption: dividends are spent as they are received and do not earn any interest. Thus, the future value of dividends received is 2 × 3 = 6. The selling price is $55. The before-tax rate of return is = (Future value of dividends + Selling Price P0 ) 1/3 - 1 = (6 +5055 )1/3- 1 = .0685 = 6.85% Annual after-tax dividends = 2 – .3492 = 1.6508 (from question 48) Future value of dividends received = 1.6508 × 3 = 4.9524 Capital gain tax = .5 × (.22 + .119) × (55 – 50) = .8475 The after-tax rate of return is 7-17 = (Future value of after-tax dividends + Selling Price - Capital Gain Tax P0 ) 1/3 - 1 = (4.9524 + 55 - .8475 50 ) 1/3 - 1 = .0573 = 5.73% 49. a. Both of these instruments are perpetuities. Recall the price of a perpetuity is P0 = annual cash flow required rate of return Rearranging the equation to find the required rate of return. The consol’s annual cash flow is its coupon payment and the preferred share’s annual cash flow is its dividend payment. Consol rate of return = bond coupon/P0 = .04 × 1000/800 = .05 = 5% Preferred share rate of return = preferred dividend/P0 = 6/120 = .05 = 5% b. Convert the annual cash flows to their after-tax amounts: After-tax bond coupon = (1 - .35) × .04 × 1000 = .65 × 40 = 26 After-tax preferred dividend = (1 - .29) × 6 = 4.26 Consol after-tax rate of return = after-tax coupon/P0 = 26/800 = .0325 = 3.25% Preferred after-tax rate of return = after-tax dividend/P0 = 4.26/120 = .0355 = 3.55% c. With a 35% corporate tax rate, the after-tax rate of return on the consol is the same as we calculated in (b), 3.25%. However, the corporate tax rate on dividends received from other Canadian corporations is zero. Thus the rate of return on the preferred shares is the before-tax rate from (a), 5%. d. Corporations with spare cash to invest will prefer to purchase dividend-paying securities of Canadian corporations than corporate bonds to take advantage of preferential dividend tax treatment. 50. The growing annuity formula is rC1 - g × [1 – ( 1 + g 1 + r ) T] Years 1 – 4 r = 12%, g = 10%, T = 4, C1 = DIV1 = (1 + g) × DIV0 = 1.1 × 1 = 1.1 Present value of Year 1 – 4 dividends = DIV1 r - g × [1 – ( 1 + g 1 + r ) T] = 1.1 .12 - .1 × [1 – ( 1.1 1.12 ) 4]= 3.82 7-18 Years 5 – 14 r = 12%, g = 8%, T = 10 To get the Year 5 dividend (which is the first cash flow in the second interval of constant growth), figure out the Year 4 dividend first. Since dividends are expected to grow 10% a year for 4 years and then grow at 8%: DIV4 = 1.14 × DIV0 = 1.14 × 1 = 1.4641 DIV5 = (1 + g) × DIV4 = 1.08 × 1.4641 = 1.581228 Present value of Years 5 – 14 dividends at the end of Year 4 = DIV5 r - g × [1 – ( 1 + g 1 + r ) 10] = 1.581228 .12 - .08 × [1 – ( 1.08 1.12 ) 10]= 12.0523 Present value of Years 5 – 14 dividends today = (1 +1 r)4 × DIV5 r - g × [1 – ( 1 + g 1 + r ) 10] = 1 1.124 × 12.0523 = 7.66 Year 15 and on r = 12%, g = 5% DIV14 = (1.08)10 × DIV4 = (1.08)10 × 1.4641 = 3.1609 DIV15 = (1.05) × DIV14 = 1.05 × 3.1609 = 3.3190 Present value of Year 15 and on dividends at the end of Year 14 = DIV15 r - g = 3.3190 .12 - .05 = 47.41 Present value of Year 15 and on dividends today =(1 +1 r)14 DIV15 r - g = 1 (1.12)14 × 47.41 = 9.70 Price today = present value of all dividends to be received P0 = 3.82 + 7.66 + 9.70 = 21.18 51. Event PV of dividends High quality gold 8 × annuity factor(9%, 20 years) = 73.03 Medium quality gold 2 × annuity factor(9%, 20 years) = 18.26 No gold 0 P0 = .4 × 73.03 + .5 × 18.26 = $38.34 52. a. Price on May 1, 2007 = 2.50/(.1 - .06) = 62.50 b. Price on May 1, 2007 = 1.248/(.1 - .04) = 20.8 7-19 c. Price on May 1, 2006 = 1 1.1 [1.20 + .3 × 62.50 + .7 × 20.8] = $31.37 d. Rate of return if R&D is successful = 1.2 + 62.50 31.37 - 1 = 1.031 = 103.1% Rate of return if R&D is unsuccessful = 1.2 + 20.8 31.37 - 1 = -0.299 = -29.9% Expected rate of return = .3 × 1.031 + .7 × (-.299) = .10 = 10% 53. The difference in rates on long- versus short-term bonds does not necessarily reflect a profit opportunity that would be a contradiction of the efficient market hypothesis. The text points out that the slope of the yield curve might be a reflection of investors’ expectations of future short-term interest rates. For example, a downward sloping yield curve, in which long-term rates are below short-term rates, might indicate that investors anticipate that future short-term rates will be lower than today’s values. Today’s long- term rates, which reflect beliefs about the future, are therefore lower than current short- term rates. 54. a. No. The split is purely a “paper transaction,” in which more shares are printed and distributed to shareholders. b. No. Operating profits continue as before the split. c. Earnings per share will fall by half. With unchanged earnings and double the number of shares, all per share values will fall by 50 percent. d. The firm’s stock price will fall by half. With unchanged total value, and double the number of shares, price per share will fall by 50 per cent. e. The shareholder’s wealth should be unaffected. Each owns double the number of shares, but the value of each share is only half of the original value. Note: These answers are actually only part of the story. Most stock split announcements are regarded as good news by investors, and stock prices generally increase. This is not because investors suffer from financial illusions, but because managers commit to splits only when they are confident that they can maintain or increase earnings. Thus the split is good news, not because it multiplies the number of shares, but because it reveals management’s confidence about the future. 55. Standard & Poor's Expected results: An opportunity to apply the ideas of the chapter to the real world. Students might not be able to see sensible patterns. The real world is not as tidy as one would like! 7-20 56. Yahoo Finance: Adobe Systems (ADBE) American Electric Power (AEP) Price Earnings Ratio: 12.15 Price Earnings Ratio: 14.59 Earnings Forecast: 2.57 Earnings Forecast: 3.23 Stock Price: 22.69 Stock Price: 37.06 PVGO (ADBE) = Price – (EPS / r) = 22.69 – (2.57 / 0.08) = -9.435 PVGO (AEP) = 37.06 – (3.23 / 0.08) = -3.315 AEP has a higher P/E ratio compared to ADBE; therefore, AEP is more similar to a growth stock. The PVGO calculated also are consistent with the P/E ratios because the PVGO for AEP is greater than the PVGO of ADBE. Investors place a higher value on the growth opportunities for AEP than they do in ADBE. (However, in this situation both values are negative). 57. Bonds Bonds are perpetual, never mature. The only cash flow is interest: Annual cash flow to bonds: Interest = $10 million/year Required rate of return on bonds = 5% Use the perpetuity formula to value the bonds: Total bond market value = annual cash flow to bonds/required rate of return = $10 million/.05 = $200 million Current bond price = total bond market value/number of bonds = $200 million/150,000 = $1,333.33 per bond Shares Shares are perpetual and cash flow never grows. The only cash flow are dividends. Dividends = dividend payout ratio × Net income Net income = (EBIT - Interest) - Corporate taxes = (70 million - 10 million) - taxes Corporate taxes = tax rate × (EBIT - Interest) = .3 × (70 million - 10 million) = 18 m Dividends = Net income = (70 million - 10 million) - 18 million = 42 million Required rate of return on equity = 11% Use the perpetuity to value the shares: Total equity value = annual cash flow to shares/required rate of return = 42 million/ .11 = $381.82 million Current share price = total equity market value/number of shares = $381.82 million/15 million =$25.45 per share 7-21 Solution to Minicase for Chapter 7 The goal is to value the company under both investment plans and to choose the better investment plan. Valuation based on past growth scenario The firm has been growing at 5% per year. Notice that book value in 2012 equals book value five years ago, at the beginning of 2008 × (1.05)5. This is calculated as (80/62.7)1/5 – 1 = .0499 = 5%. Dividends are proportional to book value and also have grown at 5% annually over the past four years, (7.7/6.3)1/4 – 1 = .0514. Dividends paid in the most recent year, 2012, were $7.7 million and they are projected to be $8 million next year, in 2013, if no changes are made. To verify that this growth in dividends is feasible, calculate the sustainable growth rate. Starting in 2013, 2/3 of earnings will be paid out as dividends, and 1/3 will be reinvested. Therefore, the sustainable growth rate as of 2013 is return on equity × plowback ratio = 15% × 1/3 = 5% Finally, the relevant discount rate for the company’s cash flows is the 11% that investors believe they can earn on similar-risk investments, not the 15% return on book equity. It is great that the company earns more on its investments than investors must earn on similar-risk investments. The share price will reflect this better performance. However, the relevant cost of capital is the opportunity cost, the return on investments of similar risk to Prairie Home Stores. The value of the firm at the end of 2012 under the assumption that the past growth is continued is therefore Value2012 = DIV2013 r – g = $8 million .11 – .05 = $133.33 million and the value per share is $133.33 million/400,000 = $333.33. Therefore, it is clear that Mr. Breezeway was correct in advising his relative not to sell for book value of $200 per share. Valuation based on rapid growth scenario If the firm reinvests all earnings in 2013, 2014, 2015 and 2016, dividends paid at the end of 2017 will reach $14 million, far greater than in the constant growth scenario. Forgoing dividends allows the company to achieve this rapid growth. 7-1 At the end of the rapid growth period, Prairie Homes will re-establish its dividend policy of paying two-thirds of its earnings as dividends. At this point, the dividend growth rate will fall to sustainable growth rate of 5%, which is the return on equity times the plowback ratio, 15% × 1/3. The value of the firm in 2016 will be the present value of the dividend in one year, $14 million, divided by the required rate of return, 11%, minus the sustainable growth rate of 5%: Value2016 = DIV2017 r – g = $14 million .11 – .05 = $233.3 million The value of the firm as of today, the end of 2012, is the present value of this amount. Remember, there will be no dividend flows in the years leading up to 2017. Value2012 = 233.3 1.114 = $153.70 million or a per share value of $153.70 million/400,000 = $384.25 Thus, it appears that the rapid growth plan is in fact preferable. If the firm follows this plan, it will be able to go public — that is, sell its shares to the public — at a higher price. One potential obstacle will be the family members who are not willing to give up their dividends. However, they should be persuaded that with an active secondary market for their shares, they will be able to sell off part of their holdings for cash. 7-2 Solution Manual for Fundamentals of Corporate Finance Richard A. Brealey, Stewart C. Myers, Alan J. Marcus, Elizabeth Maynes, Devashis Mitra 9780071320573, 9781259272011
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